Acceleration of the Power Method with Dynamic Mode Decomposition

Abstract

Presented is an algorithm based on dynamic mode decomposition (DMD) for acceleration of the power method (PM). The power method is a simple technique for determining the dominant eigenmode of an operator $\mathbf{A}$, and variants of the power method are widely used in reactor analysis. Dynamic mode decomposition is an algorithm for decomposing a time-series of spatially-dependent data and producing an explicit-in-time reconstruction for that data. By viewing successive power-method iterates as snapshots of a time-varying system tending toward a steady state, DMD can be used to predict that steady state using (a sometime surprisingly small) $n$ iterates. The process of generating snapshots with the power method and extrapolating forward with DMD can be repeated. The resulting restarted, DMD-accelerated power method (or DMD-PM($n$)) was applied to the two-dimensional IAEA diffusion benchmark and compared to the unaccelerated power method and Arnoldi's method. Results indicate that DMD-PM($n$) can reduce the number of power iterations required by approximately a factor of 5. However, Arnoldi's method always outperformed DMD-PM($n$) for an equivalent number of matrix-vector products $\mathbf{Av}$. In other words, DMD-PM($n$) cannot compete with leading eigensolvers if one is not limited to snapshots produced by the power method. Contrarily, DMD-PM($n$) can be readily applied as a post process to existing power-method applications for which Arnoldi and similar methods are not directly applicable. A slight variation of the method was also found to produce reasonable approximations to the first and second harmonics without substantially affecting convergence of the dominant mode.